Metamath Proof Explorer

Theorem seglecgr12

Description: Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion seglecgr12 ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) ) )

Proof

Step Hyp Ref Expression
1 df-3an ⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) ↔ ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) )
2 seglecgr12im ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) → ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) )
3 1 2 syl5bir ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) → ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) )
4 3 expd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ → ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) ) )
5 simp11 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
6 simp12 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
7 simp13 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
8 simp23 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
9 simp31 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) )
10 cgrcom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ↔ ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ) )
11 5 6 7 8 9 10 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ↔ ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ) )
12 simp21 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
13 simp22 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
14 simp32 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) )
15 simp33 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) )
16 cgrcom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ↔ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
17 5 12 13 14 15 16 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ↔ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
18 11 17 anbi12d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) ↔ ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) )
19 df-3an ⊢ ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) ↔ ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) )
20 seglecgr12im ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) → ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) )
21 5 8 9 14 15 6 7 12 13 20 syl333anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) → ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) )
22 19 21 syl5bir ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) → ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) )
23 22 expd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐸 , 𝐹 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐺 , 𝐻 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) → ( ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ → ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) ) )
24 18 23 sylbid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) → ( ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ → ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ) ) )
25 4 24 impbidd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐺 , 𝐻 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐸 , 𝐹 ⟩ Seg≤ ⟨ 𝐺 , 𝐻 ⟩ ) ) )